In the special case that \(Z\)  is a one dimensional unitary character \(\mathbb {C}_{\lambda }\)  of \(L\) , we denote \(\mathcal {L}_S(Z)\)  as \(A_{\mathfrak {q}}(\lambda )\) . This module has infinitesimal character \(\lambda +\rho ^\prime \) . As shown by Salamnaca-Riba in [1]}, any irreducible unitary \((\mathfrak {g}, K)\)  module \(\pi \)  with a strongly regular real infinitesimal character \(\Lambda \)  must be isomorphic to an \(A_{\mathfrak {q}}(\lambda )\)  module. Here \(\Lambda \)  being strongly regular means that
\(B(\Lambda -\rho ^\prime , \alpha )\ge 0, \quad \forall \alpha \in (\Delta ^+)^\prime (\mathfrak {g}, \mathfrak {t}_f).\) 
